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We don't have a direct method of finding the Modulus, but we can proceed as follows:

If the Modulus =M If the quotient =Q for the first and =q for the second, then we have: MQ + 199 = 57,131.................(1), and: Mq + 67 =37,139...................(2) Will ignore the Modulus "M" for now and re-write the two equations as: Q =57,131 - 199 =56,932.........(3) q =37,139 - 67 =37,072.........(4). Will factor (3) and (4) as follows: 56,932 = 2^2 * 43 * 331, and: 37,072 = 2^4 * 7 * 331 From the above factorization, we can readily see that the biggest factor they have in common is =331. Then we have: 57,131 mod 331 = 199, and 37,139 mod 331 = 67. And that satisfies both equations.

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